We investigate equational characterizations of bisimulation and trace metaequivalences defined on the ULTraS metamodel. This yields general axiomatizations encompassing those appeared in the literature, as well as new ones, for bisimulation and trace equivalences defined on specific classes of processes. The results are provided incrementally, by starting with the core axioms and then singling out additional axioms for bisimulation post- and pre-metaequivalences on the one hand, and different additional axioms for trace post- and pre-metaequivalences on the other hand. The various axioms highlight the fundamental differences in the discriminating power of the four metaequivalences, regardless of specific classes of processes.